An introduction to Rota s universal operators: properties, old and new examples and future issues

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1 Concr. Oper. 216; 3: Concrete Operators Open Access Research Article Carl C. Cowen and Eva A. Gallardo-Gutiérrez* An introduction to Rota s universal operators: properties, old and new examples and uture issues DOI /conop Received December 3, 215; accepted January 26, 216. Abstract: The Invariant Subspace Problem or Hilbert spaces is a long-standing question and the use o universal operators in the sense o Rota has been an important tool or studying such important problem. In this survey, we ocus on Rota s universal operators, pointing out their main properties and exhibiting some old and recent examples. Keywords: Rota s universal operators, Invariant subspace, Analytic Toeplitz operator, Lomonosov s Theorem MSC: 47B99 1 Preace The Invariant Subspace Problem is a remarkable open problem which asks i every bounded, linear operator acting on a separable, complex Hilbert space has a non-trivial closed invariant subspace. In order to provide an answer, operator theorists have developed, over more than six decades, dierent approaches connecting areas like Function Theory or Measure Theory to general Functional Analysis. We are indebted to Proessors Niccola Arcozzi and Isabelle Chalendar as they invited us to give a mini course on Universal Operators at the conerence XII Advanced Course in Operator Theory and Complex Analysis held in Bologna, Italy, in June 215. These notes, which survey some o the topics addressed in the mini course, consist o an elementary account o certain aspects o the theory o universal operators; the choice o topics was naturally heavily inluenced by our own background. We start by introducing some basic properties and derive some consequences in the study o the invariant subspaces o an operator. These notes are not a survey o dierent approaches to solve the Invariant Subspace Problem; hence, many important aspects and the various connections to other questions have been omitted. Similarly, the bibliography is ar rom being complete. This is compensated by the existence o two excellent books, a classical one due to Radjavi and Rosenthal [24], and a recent one due to Chalendar and Partington [3]. In act, the present notes could be seen as a (hopeully useul) preparatory material or the reader who wants to study Universal Operators" in more detail. Besides Proessors Arcozzi and Chalendar, we want to thank all the participants o the Bologna conerence or a very enjoyable meeting. Carl C. Cowen: Indiana University-Purdue University Indianapolis, USA, ccowen@iupui.edu *Corresponding Author: Eva A. Gallardo-Gutiérrez: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid e ICMAT, Plaza de Ciencias 3, 284, Madrid, Spain, eva.gallardo@mat.ucm.es 216 Cowen and Gallardo-Gutiérrez, published by De Gruyter Open. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3. License.

2 44 C.C. Cowen, E.A. Gallardo-Gutiérrez 2 Introduction: A special class o operators In 196, Rota [27] introduced the idea o an operator whose lattice o invariant subspaces has a structure rich enough to model every Hilbert space operator and showed, perhaps surprisingly, that such operators exist. Throughout these notes, all subspaces are assumed to be closed. Deinition 2.1 ([3, p. 213]). Let H be a Hilbert space, U a bounded operator on H and B.H/ the algebra o bounded operators on H. We say U is universal or H i or each non-zero bounded operator A on H there is an invariant subspace M or U and a non-zero number such that A is similar to U j M, that is, there is a linear isomorphism X o H onto M such that UX D XA. Now, A and A have the same invariant subspaces and the similarity X takes invariant subspaces o A to invariant subspaces o U j M. Suppose U is a universal operator or a separable, ininite dimensional Hilbert space H. Then every bounded operator on H has an invariant subspace i and only i every ininite dimensional subspace M o H that is invariant or U has a non-zero, proper subspace M that is also invariant or U. In other words, understanding the invariant subspace problem on Hilbert spaces becomes a question o understanding the invariant subspaces o the single operator U. In 1969 Caradus showed a suicient condition to ensure that a given operator in a Hilbert space is universal. The best known examples o universal operators, including the operator Rota used to introduce the concept, satisy the hypotheses o Caradus Theorem. Theorem 2.2 ([2, p. 527] or see [3, p. 214]). I H is a separable Hilbert space and U is a bounded operator on H such that: 1. the null space o U is ininite dimensional, 2. and the range o U is H, then U is universal or H. The best known example o a universal operator on a separable Hilbert space H is the adjoint o a unilateral shit o ininite multiplicity. Such an example was introduced by Rota in 196 [27], and it can be regarded as S acting on 1X `2.H/ D.h j / j W k.h j / j k 2 D kh j k 2 H < 1g j D by S..h ; h 1 ; h 2 ; // D.h 1 ; h 2 ; /: Clearly, S satisies both hypotheses o the Caradus Theorem. Indeed, let us point out that any universal operator U must satisy the irst hypothesis o the Caradus Theorem because it models operators with ininite dimensional kernels. Nevertheless, surjectivity is not necessary, though there must exist a closed invariant subspace M o U such that U j M has ininite dimensional kernel and it is surjective (once again, because it models such operators). Hence, an operator U is universal in a separable Hilbert space H i and only i there exists a closed invariant subspace M o U such that U j M satisies Caradus Theorem. The next result provides a version o the Caradus Theorem with a broader conclusion. Theorem 2.3 ([9]). I H is a separable Hilbert space and U is a bounded operator on H such that: 1. the null space o U is ininite dimensional, 2. and the range o U is H, then there is > so that or jj <, the operator U C I is universal. Moreover, or any complex number and any bounded operator A on H, there is an invariant subspace M or U and constants and ˇ such that A D.U C I /j M is similar to A C ˇI. In particular, the lattices o invariant subspaces or A and A are isomorphic as lattices. The proo, taken rom [9, pp. 487], is included or the sake o completeness.

3 An introduction to Rota s universal operators 45 Proo. Let e n g n2n be an orthonormal basis or the Hilbert space H and let en g n2n be an orthonormal basis or N D v W Uv D g, the kernel o U. Deining W on H by W e n D en or each positive integer n and extending linearly means W is isometric on H and U W D. Since the range o U is H, the restriction o U to N? is an invertible operator and we let V be its inverse. That is, V W H 7 N? and U V D I. Caradus showed that i T is an operator on H or which kt k < kv k 1, then X D P 1 kd V k W T k is a bounded operator with closed range, the range o X is invariant or U, and UX D XT, that is, X gives a similarity between T and the restriction o U to its invariant subspace, range.x/. Using the notation above, let D kv k 1. Given a complex number with jj < and a bounded operator A on H, we want to ind a constant so that there is an invariant subspace M or U CI so that A D.U C I /j M is similar to A, which will mean U CI is universal. By our choice o, we see jj is positive and we choose so that ka k C jj <. This means that T D A I satisies kt k ka k C jj < kv k 1. Thus, the construction o Caradus shows X D P 1 kd V k W T k is a bounded operator on H with closed range, the range o X is invariant or U, and X gives a similarity between the restriction o U to its invariant subspace M D range.x/ and the operator T D A I. Now M is also invariant or U C I and.u C I /j M T C I D A Since A was a arbitrary operator on H and we have shown that there is an invariant subspace M so that.u C I /j M is similar to a multiple o A, we see that U C I is a universal operator, as we were to prove. The inal conclusion ollows rom the universality o U and the act that the lattices o U and U C I are the same. 3 Examples o universal operators As we already pointed out, the best known example o a universal operator on a separable Hilbert space H is the adjoint o a unilateral shit o ininite multiplicity. Since Rota showed that such an operator is universal (see [27]), there have been more examples in the same vein. For instance, i a denotes a positive number, a >, and L 2.; 1/ is the space o Lebesgue measurable unctions on on.; 1/ such that R 1 j j2 dx < 1, the operator T a.t/ D.t C a/; or t > ; is universal. This is a consequence o the act that T a acting on L 2.; 1/ is similar to the adjoint o the shit o ininite multiplicity (see [3, Chapter 8], or instance). Another interesting class o examples, which relies on the same idea, consists o the adjoints o analytic Toeplitz operators induced by singular unctions acting on the classical Hardy space. More precisely, let D denote the open unit disc in the complex plane and H 2 the classical Hardy space: H 2 D h analytic in D W sup Z 2 <r<1 jh.re i /j 2 d 2 < 1g: For a bounded analytic unction on the unit disk, that is, is in H 1, the analytic Toeplitz operator, T, on H 2 is the operator deined by.t h/.z/ D.z/h.z/ or h in H 2. For in H 1, the operator T is bounded on H 2 and it is easy to prove that kt k D k k 1. More generally, i is a unction in L 1.@D/, the Toeplitz operator T is the operator on H 2 given by T h D P C h where P C is the orthogonal projection rom L 2.@D/ onto H 2 and h is a unction in H 2. Also when is in L 1.@D/, the operator T is bounded on H 2 and kt k D k k 1. In the case that is in H 1, the projection P C has no eect: or h in H 2 and in H 1, P C h D h. Douglas s book [1] can provide some background on properties o Toeplitz operators. The ollowing lemma is important in order to ensure that many analytic Toeplitz operators have universal adjoints.

4 46 C.C. Cowen, E.A. Gallardo-Gutiérrez Lemma 3.1 ([9]). I is a unction in H 1 and there is ` > so that j.e i /j ` almost everywhere on the unit circle, then 1= is in L 1.@D/ and the (non-analytic) Toeplitz operator T 1= is a let inverse or the analytic Toeplitz operator T. With that lemma at hand, the ollowing theorem can be proved by means o Caradus Theorem. Theorem 3.2 ([9]). Let 2 H 1 such that 1= 2 L 1.@D/. I the Toeplitz operator T dimensional kernel, then T is universal in H 2. has ininite Another well-known example o a universal operator was presented by Nordgren, Rosenthal and Wintrobe in the eighties (see [21, 22]). They proved that i ' is a hyperbolic automorphism o the unit disc and is in the interior o the spectrum o the composition operator C ' acting on H 2, then C ' I is a universal operator on H 2. O course, the lattices o the closed invariant subspaces o C ' I and C ' coincide, so they have the same the minimal invariant subspaces. Observe that given a unction 2 H 2 the minimal closed invariant subspace or C ' that contains is, precisely, the closure (in H 2 ) o the linear span generated by the orbit o, that is, spanc n ' W n gh 2 : Invariant subspaces generated by the orbit o a given unction are commonly called cyclic subspaces (see [24]). In [14], Gallardo-Gutiérrez and Gorkin studied the behavior o the unctions in the Hardy space in order to determine when the cyclic subspaces generated by them under C ' are minimal. Observe that inding a minimal invariant subspace o dimension strictly bigger than 1 or C ' (and thereore, ininite dimensional) would imply the existence o a bounded, linear operator without non-trivial, closed invariant subspaces in a Hilbert space; and thereore the answer to the Invariant Subspace Problem. In this context, with the goal o a better understanding o the lattice o the invariant subspaces o C ', Cowen and Gallardo-Gutiérrez in [7] showed, among other things, that C ' is similar to the adjoint o an analytic Toeplitz operator T? whose symbol is a covering map o an annulus, behaving, in some sense, like the adjoint o the shit o ininite multiplicity. This direction was also taking urther in [23], where Partington and Pozzi provided more examples o universal weighted shit operators. As a inal comment to this Section, we point out the common property shared by most o the examples o universal operators presented up to now: they are either adjoints o shit operators o ininite multiplicity or similar to restrictions o them to invariant subspaces. 4 Universal operators commuting with compact operators In 1973, Lomonosov [17] proved a remarkable theorem that probably, up to this point, is the main airmative result in the context o the Invariant Subspace Problem or operators on general separable, complex Banach spaces: any linear bounded operator T, not a multiple o the identity, has a nontrivial invariant closed subspace i it commutes with a non-scalar operator that commutes with a nonzero compact operator. For some years, looking or an operator not satisying the hypotheses o Lomonosov s Theorem in the context o Hilbert spaces was an important goal. Finally, in 198, Hadwin, Nordgren, Radjavi and Rosenthal provided such an operator (see [15]). In the context o Banach spaces, Enlo showed the existence o a separable, complex Banach space and a linear bounded linear operator T without nontrivial closed invariant subspaces. Enlo s construction was ingenious and very diicult; the main idea was to start with the operator o multiplication by the independent variable on the space o polynomials P and construct a norm on the space so that every non-zero vector is cyclic or the extension o the operator to the completion o the polynomials (see [13]). In 1985, Read [25] constructed a bounded linear operator without nontrivial closed invariant subspaces in the well-known sequence space `1. His construction, which appears to be the irst example o such an operator on any o the classical Banach spaces, was simpler than Enlo s in some sense. Read, subsequently, made an even more remarkable construction o a bounded linear operator on `1 that has no closed invariant sets except the trivial ones (see [26]).

5 An introduction to Rota s universal operators 47 Regarding universal operators in the context o Lomonosov s Theorem, let us remark that one o the common properties o the examples in the previous section is that they do not commute with an injective compact operator with dense range. Indeed, in [4, Thm. 1] and [5], it is shown that neither the Nordgren, Rosenthal and Wintrobe operator nor the adjoint o a unilateral shit o ininite multiplicity can commute with a nontrivial compact operator. Let us analyze the case in which S is an analytic Toeplitz operator on the Hardy space H 2 whose symbol is a singular inner unction or ininite Blaschke product. First, let us recall that an inner unction is an analytic unction in D with contractive values, i.e., j.z/j 1 or z 2 D, such that its boundary values.e it / D lim r1.reit / (which exist or almost every point e it with respect to Lebesgue measure on the unit circle) have modulus one or almost all e it. A classical result that traces back to Beurling [1] in 1948 states that every inner unction can be actorized as product, in principle, o two inner unctions: one collecting all the zeros o in D (a Blaschke product) and the other, lacking zeros in D, a singular inner unction, i.e., it can be expressed by an explicit integral ormula in terms o a singular measure on the unit circle. More precisely, i D BS where B is a Blaschke product having a zero o multiplicity N at, then we may write B.z/ D z N Y j 1 z j jz j j z z j 1 z j z ; where jj D 1, z j 2 D n g and P j 1.1 jz j j/ < 1. For the second actor, S, there is a constant o modulus one and a positive inite measure,, singular with respect to Lebesgue measure, such that 1 Z2 S.z/ D e it C z e it z d.t/ A : We reer to the classical books [12] or [16] or more on inner unctions. Now, i S is an analytic Toeplitz operator on the Hardy space H 2 whose symbol is a singular inner unction or ininite Blaschke product, it is straightorward that S is an isometric operator and S has ininite dimensional kernel mapping H 2 onto H 2. Caradus Theorem [2] yields that S is a universal operator. Using the Wold Decomposition Theorem (see the classical book [18], or instance), such an operator can be represented as a block matrix on H D 1 kd S k W where W D H 2 SH 2 that is upper triangular and has the identity on the super-diagonal: 1 I S I B I A : :: An easy computation shows that every operator that commutes with S has the orm 1 A A 1 A 2 A 3 A A 1 A 2 A B A 1 C A : :: an upper triangular block Toeplitz matrix, that is, an upper triangular block matrix whose entries on each diagonal are the same operator on the ininite dimensional Hilbert space W. Because every block in such a matrix occurs ininitely oten, it is easy to see that the only compact operator that commutes with the universal operator S is, not an interesting compact operator. In [8] the authors proved the ollowing: Theorem 4.1 ([8]). There exists a universal operator or separable, ininite dimensional complex Hilbert spaces that commutes with an injective compact operator with dense range.

6 48 C.C. Cowen, E.A. Gallardo-Gutiérrez In order to prove the theorem, the authors exhibited a domain C such that the covering map o such a domain induces an analytic Toeplitz operator with universal adjoint. Moreover, they showed the existence o a weighted composition operator W ;J D T C J in H 2, with both the analytic Toeplitz T and the composition C J bounded operators, such that W ;J is an injective compact operator with dense range that commutes with the universal operator T. Observe that once we have such a compact operator in the commutant o T, denoted by T g, we have a whole subalgebra. More precisely, i K denotes the (non-empty) set o compact operators that commute with T, that is, K D G 2 B.H 2 / W G is compact, and T G D GT g then, we may state the ollowing Theorem 4.2 ([9]). The set K is a closed subalgebra o T g that is a two-sided ideal in T g. In particular, i G is a compact operator in K and g and h are bounded analytic unctions on the disk, then T g G, GT h, and T g GT h are all in K. Moreover, every operator in K is quasi-nilpotent. As a consequence o Theorem 4.1, it is possible to describe invariant linear maniolds (not necessarily closed) or any bounded linear operator acting on a separable, ininite dimensional Hilbert space. In order to do that, let A be a bounded linear operator in H and let M be a closed invariant subspace o T such that the restriction o T to M is similar to A. We may assume, without loss o generality, that A is indeed the restriction o T to M. First, we observe that M has ininite codimension. Indeed, such a property ollows rom the ollowing general statement regarding invariant subspaces o adjoints o analytic Toeplitz operators in the Hardy space. Proposition 4.3 ([9]). I is a non-constant bounded analytic unction and M is a proper invariant subspace or T, then M? is ininite dimensional. We include the proo, taken rom [9, p. 492], or the sake o completeness. Proo. We have assumed that M H 2, so M?./. Since M is invariant or T, the subspace M? is invariant or.t / D T. Now T is an analytic Toeplitz operator with non-constant, so T has no eigenvalues. This means the restriction o T to its invariant subspace M? also has no eigenvalues. But every operator on a inite dimensional space has eigenvalues, so M? must be ininite dimensional. Let us write H 2 as the direct sum o M with its orthogonal complement, H 2 D M M?. This allows us to give a block representation o T with respect to this splitting. Since M is invariant or T, the block matrix or T is upper triangular. We will denote the operators T D T and W D W ;J by the block matrices: T D T A B and W D.T C J / P Q (1) C R S The act that T D T these two matrices: and W D W ;J W T T W commute gives inormation about the interactions between the entries o P Q R S A B C A B D C P Q D R S PA PB C QC RA RB C SC AP C BR AQ C BS CR CS Equating these two computations, we see PA D AP C BR (2) RA D CR (3) Since A is the operator o primary interest, Equation (2) is not so interesting i P D. The ollowing lemma says we can always avoid this situation by replacing T and W ;J by et and ew where P is not, but at the cost o

7 An introduction to Rota s universal operators 49 having T and W being similar to, but not necessarily being, adjoints o an analytic Toeplitz operator and weighted composition operator, respectively (see [8] or the proo). Lemma 4.4 ([8]). I the universal operator T D T and the compact operator W D W ;J have the representations o Equation (1) then there are a universal operator et and an injective compact operator ew with dense range that commute or which ep in a replacement o P in a new version o Equation (1) is not zero, that is, without loss o generality, we may assume P. The ollowing two results are consequences o the description above. For the irst one, we recall that a closed subspace L on a Hilbert space H is said to be a nontrivial hyperinvariant subspace or the bounded operator F i L./, L H, and L is invariant or every operator that commutes with F. Lomonosov s Theorem [17] can be stated as Non-scalar operators that commute with a nontrivial compact operator have hyperinvariant subspaces. Theorem 4.5 ([8]). Let the universal operator T and the commuting injective compact operator W with dense range have the representations o Equation (1) with P. Then the ollowing are true: [I] Either R or A has a nontrivial hyperinvariant subspace. [II] Either ker.r/ D./ or A has a nontrivial invariant subspace. [III]Either B or A has a nontrivial hyperinvariant subspace. The second consequence is the description o invariant linear maniolds (not necessarily closed) or A as ollows Corollary 4.6. [9] Since RA D CR, we have R C D A R and i L is any invariant subspace or C then R L is an invariant linear maniold or A. Proo. Suppose v is a vector in R L, say v D R w or w in L. Then A v D A R w D R.C w/ which is in R L because C w is in L. Thus, R L is an invariant linear maniold o A. Note that this corollary does not make any assertion about the size o R L, but i R L is dense or i R L D./, this is not helpul in order to provide closed invariant subspaces or A. 5 Universal Toeplitz operators and subspaces invariant or the backward shit In this last section, our aim is to point out a connection between the invariant subspaces o the shit operator and the lattice o invariant subspaces o Toeplitz operators having universal adjoints. First, let us recall that one o the most remarkable results in the study o invariant subspaces o shit operators is the celebrated Beurling Theorem. Proved in 1949 by Beurling [1], it states that any shit-invariant subspace has the orm H 2, where is an inner unction. As a consequence o Lomonosov s Theorem, we get the ollowing result related to commutants o analytic Toeplitz operators. Theorem 5.1 ([9]). I is a non-constant bounded analytic unction or which T commutes with a non-zero compact operator, there is a backward shit invariant subspace, L D.H 2 /? or some inner unction, that is invariant or every operator in the commutant T g. Observe that i T is a universal operator on H 2 in particular, it commutes with T or every inner unction. Since the kernels o the operators T are the vectors in H 2 that are not cyclic vectors or the backward shit, one might ask the question i every closed, ininite dimensional subspace o H 2 includes a non-zero, non-cyclic vector or the backward shit. The cyclic and non-cyclic vectors or the backward shit acting on H p, 1 < p < 1, were characterized by Douglas, Shapiro, and Shields [11] in terms o pseudocontinuation. Moreover, they showed that the set o non-cyclic vectors is a (not closed) linear maniold in H p. Such a question has a negative answer, as we

8 5 C.C. Cowen, E.A. Gallardo-Gutiérrez were inormed by Pro. N. Nikolski [2]. Indeed, it is possible to construct ininite dimensional, closed subspaces consisting only o cyclic vectors or the backward shit (see the example in [19, pp. 83] and the one included in [9]). Moreover, in [9], the authors present a urther discussion o non-cyclic vectors or the backward shit in relation to analytic Toeplitz operators with universal adjoints. In particular, they introduce the ollowing deinition: Deinition 5.2. I M is a closed subspace o H 2 and v is a non-zero vector in M, we say v is a sharp vector or M i v is non-zero and not a cyclic vector or T z and the smallest invariant subspace o T z that contains v does not contain all o M. Indeed, v is a sharp vector or M i M \ kernel.t / M or some inner unction. This gives the ollowing result: Theorem 5.3 ([9]). Let T be the adjoint o analytic Toeplitz operator that is universal on H 2 and let M be an ininite dimensional, proper invariant subspace or T. I is any inner unction, then M \kernel.t / is a subspace o M that is invariant or T. I there is a non-zero vector in M \kernel.t / and M is not contained in kernel.t /, then M \ kernel.t / is a proper subspace o M that is invariant or T. As a consequence we may prove the ollowing result regarding the Invariant Subspace Problem. Corollary 5.4 ([9]). Let T be the adjoint o analytic Toeplitz operator that is universal on H 2 and let M be an ininite dimensional, proper invariant subspace or T. 1. I M contains a cyclic vector or the backward shit, T z, and a non-zero vector that is not cyclic or the backward shit, then there is a proper subspace o M that is invariant or T 2. I M is an invariant subspace or the backward shit, T z, then there is a proper subspace o M that is invariant or T. I every ininite dimensional, proper invariant subspace, M, or T linear operator in a Hilbert space has a non-trivial closed invariant subspace.. must satisy either (1) or (2), then any bounded Finally, let us point out that we particularly believe that these results provide a wide variety o tools or the study o the Invariant Subspace Problem and that choices can be made between dierent kinds o universal operators based on the types o questions under study. Acknowledgement: The authors are grateul to the reeree or a careul reading as well as constructive suggestions which improved signiicantly the submitted version o this manuscript. This work is partially supported by Plan Nacional I+D grant no. MTM P. This work was supported by the LABEX MILYON (ANR-1-LABX-7) o Université de Lyon, within the program Investissements d Avenir (ANR-11-IDEX-7) operated by the French National Research Agency (ANR). Reerences [1] A. Beurling, On two problems concerning linear transormations in Hilbert space, Acta Math. 81(1949), [2] S. R. Caradus, Universal operators and invariant subspaces, Proc. Amer. Math. Soc. 23(1969), [3] I. Chalendar and J. R. Partington, Modern Approaches to the Invariant Subspace Problem, Cambridge University Press, 211. [4] C. C. Cowen, The commutant o an analytic Toeplitz operator, Trans. Amer. Math. Soc. 239(1978), [5] C. C. Cowen, The commutant o an analytic Toeplitz operator, II, Indiana Math. J. 29(198), [6] C. C. Cowen, An analytic Toeplitz operator that commutes with a compact operator and a related class o Toeplitz operators, J. Functional Analysis 36(198), [7] C. C. Cowen and E. A. Gallardo-Gutiérrez, Unitary equivalence o one-parameter groups o Toeplitz and composition operators, J. Functional Analysis 261(211), [8] C. C. Cowen and E. A. Gallardo-Gutiérrez, Rota s universal operators and invariant subspaces in Hilbert spaces, to appear.

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